Equation of Motion from Action for a Scalar Field + Matter

In a review on quintessence, the equations of motion (EoM) for the action
$$
S=int!mathrm{d}^4xsqrt{-g}left(frac{M_p^2R}{2}-frac{g^{munu}partial_muphipartial_nuphi }{2}-Vleft(phiright)right)+S_m,
$$
are given by
$$
3M_P^2H^2=dot{phi}^2/2+V(phi)+rho_m
$$
and
$$
2M_p^2dot{H}=-[dot{phi}^2+(1+omega_m)rho_m].
$$
I don’t really see, how?! There has to be some assumption on $S_m$, I assume, otherwise there wouldn’t be the $omega_m$ and $rho_m$ terms. Or could they be introduced through any general relation?

$M_p$ is the Planck mass, $R$ the Ricci scalar, $g$ the determinant of the metric tensor, $phi$ a scalar field, $V$ its potential, $S_m$ the matter action, $H$ the Hubble parameter, and $p=omegarho$ the equation of state.